How can I integrate Sqrt[x^2-a] using trig substitutions?

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SUMMARY

The integration of Sqrt[x^2 - a] can be effectively approached using trigonometric substitutions, specifically secant or tangent, depending on the sign of 'a'. For a > 0, the substitution x = sqrt(a) sec(y) is appropriate, while for a < 0, the integrand transforms to Sqrt[x^2 + b] where b = -a. The discussion emphasizes that the integral remains valid regardless of the sign of 'a', and the presence of a tangent function in the result is a common outcome when applying these substitutions.

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Could someone show me how to integrate this. Bear in mind 'a' could be positive or negative thus i don't think we can use sqrt(a) in our answer...
 
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Well if a>0, then you can have sqrt(a) in your answer...

But for a<0 => -a>0 meaning that your integrand will become sqrt{x^2+b} where b=-a
 
Shouldn't matter if it's positive or negative, the integral is still the same, imaginary or not.
 
use trig substitutions, sec or tan depending on whether a is positive or negative.
 
mathwonk said:
use trig substitutions, sec or tan depending on whether a is positive or negative.


only problem is say if I use the substitution Let x = sqrt(a)Secy the integrated result has a tany in it
 

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